Abstract: Growing theoretical and experimental research on action and language processing and on number learning and gestures clearly demonstrates the role of embodiment in cognition and language processing. In psychology and neuroscience, this evidence constitutes the basis of embodied cognition, also known as grounded cognition (Pezzulo et al. 2012; Borghi & Cangelosi 2014). In robotics, these studies have important implications for the design of linguistic capabilities in cognitive agents and robots for human-robot communication, and have led to the new interdisciplinary approach of Developmental Robotics (Cangelosi & Schlesinger 2015). During the talk we will present examples of developmental robotics models and experimental results from iCub experiments on the embodiment biases in early word acquisition and grammar learning (Morse et al. 2015; Morse & Cangelosi 2017) and experiments on pointing gestures and finger counting for number learning (De La Cruz et al. 2014). We will then present a novel developmental robotics model, and experiments, on Theory of Mind and its use for autonomous trust behavior in robots. The implications for the use of such embodied approaches for embodied cognition in AI and cognitive sciences, and for robot companion applications will also be discussed.